CATEGORICAL LOGIC
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
A Bachelor’s degree in Mathematics (or equivalent mathematical maturity). Although some familiarity with the language of category theory would be desirable, no previous knowledge of these subjects is required. Indeed, the course will present all the relevant preliminaries as they are needed.
The student will be able to choose between two alternative exam modes:
- Solutions (prepared at home) to exercises assigned by the Lecturer and seminar on a suitable topic extending the contents of the course (chosen in agreement with the Lecturer). The evaluation of this seminar will constitute the two thirds of the final grade, while the remaining third will be determined by the solutions to the exercises.
- Taking two partial written exams (each of which consisting in theoretical questions and exercises), one administered half way through the course and the other immediately after the end of it.
his course is an introduction to categorical logic, aimed at providing the student with an extensive theoretical preparation in this field both from a theoretical viewpoint and at the level of methodologies for effectively applying such techniques in a great variety of different mathematical contexts.
At the end of the course, the student
- will have learned the central notions and results of categorical logic
- will have become familiar with the interpretation of logic in categories, the development of a functorial model theory, and the theory of classifying toposes
- will know about methods for studying mathematical theories from a categorial and topos-theoretic perspective
- will have acquired techniques for establishing new fruitful connections between different fields through the use of methods of logical, categorial and topos-theoretic nature.
The study of categorical models of mathematical theories is a theme of great relevance for its applications to various sectors of mathematics and computer science. It allows the development of a functorial model theory and the study of the fundamental invariants of mathematical theories through 'logical' categories associated with them, such as their classifying toposes.
The course will provide an introduction to first-order categorical logic, after having presented the necessary preliminaries of categorical and topos-theoretic nature. It will present techniques useful for studying a given theory from a multiplicity of different points of view, and for establishing new connections between different fields of mathematics.
Particular attention will be devoted to the study of classifying toposes, that is of Grothendieck toposes from a logical point of view. It will be shown how essential properties of a mathematical theory are reflected in the properties of its classifying topos and, reciprocally, how the study of topos-theoretic invariants can lead to the introduction of significant logical notions and constructions. Through the discussion of numerous examples, we will also illustrate the sense in which Grothendieck toposes can effectively serve as 'bridges' to transfer knowledge between different mathematical theories with a common semantics.
PROGRAMME:
Categorical preliminaries
The interpretation of logic in categories
Syntactic categories
Sites and theories
Grothendieck toposes
Geometric morphisms
Morita-equivalence between theories
Classifying toposes and the 'bridge' technique
Theories of presheaf type
Subtoposes and quotients
Quotients of a theory of presheaf type
Examples and applications
F. Borceux, Handbook of categorical algebra, Vols. 1-2-3, Cambridge University Press, 1994
O. Caramello, Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’, Oxford University Press, 2017
R. Goldblatt, Topoi: The Categorial Analysis of Logic, Reprint of the 1983 edition, Dover.
P. T. Johnstone, Sketches of an Elephant: a topos theory compendium. Vols. 1-2, Oxford University Press, 2002
S. Mac Lane, Categories for the working mathematician, Springer, 2nd edition, 1997
S. Mac Lane, I. Moerdijk, Sheaves in geometry and logic. A first introduction to topos theory, corrected reprint of the 1992 edition, Universitext, Springer-Verlag, New York, 1994
M. Makkai and G. Reyes, First Order Categorical Logic, Springer, 1977
E. Riehl, Category theory in context, Cambridge University Press, 2016
The frontal theoretical lessons, given with the support of slides, will be supplemented by sessions of exercises assigned by the Lecturer in the previous lessons, in which the students who desire to do so will be able to present their solutions and discuss them with the Lecturer.
The course will consist of six hours per week, of which one or two will be devoted to the discussion of exercises or specific problems.
The students of the course can reach the lecturer in her office in the hour immediately following the end of each lecture to ask for more explanations, clarifications or suggestions for further study.
Students can also take an appointment with the Lecturer by sending her an e-mail.
Professors
Borrowers
-
Degree course in: MATHEMATICS